288 APPENDIX. 



p_ 10. 



is reduced to 



^ = KdT KLtdq + \KMt~ N PI de, 



smv 



observing that d t = dT, if T denotes the time of perihelion passage. 

 If we differentiate the equation 



r== 



1 -f- e cos v 



we find 



r , I 2 o 2 sin 2 i v 7 , r 2 e sin v , 



dr = - do -\ -- ., . de-\ 7 - r dv. 



2 r 



q 



These formulas are given by NlCOLAI, (Monatliche Correspondenz, Vol. XXVIL, 

 p. 212). The labor of using them is greatly abridged by the fact that K, L, 

 M, etc., are computed once for all, and that the quantities needed for this pur 

 pose are those required for computing the true anomaly and radius vector. 



If the ellipse so nearly approaches the parabola that, in the coefficients, we 

 may assume 



tan kv = y tan | w 



,~ _ k Y/ 2 cos 2 v 

 2 q% tan $ v 



the values of dv and dr assume a much more simple form. In this case we 

 should have 



if s j n v _ ^ y/ 2 cos 8 \v sin \ v _ k ^ 2 cos 4 v _ k \j 2 q 



(1 



and consequently, 



sn 



8] - 



+9 e) 



